6.12.32 problem 39

Internal problem ID [1886]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 05:20:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{5}+1\right ) y^{\prime \prime }+14 x^{4} y^{\prime }+10 x^{3} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

Order:=6; 
ode:=(2*x^5+1)*diff(diff(y(x),x),x)+14*x^4*diff(y(x),x)+10*x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{5}}{2}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 20

ode=(1+2*x^5)*D[y[x],{x,2}]+14*x^4*D[y[x],x]+10*x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {x^5}{2}\right )+c_2 x \]

✓ Sympy. Time used: 0.306 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(14*x**4*Derivative(y(x), x) + 10*x**3*y(x) + (2*x**5 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{5}}{2}\right ) + C_{1} x \left (1 - \frac {4 x^{5}}{5}\right ) + O\left (x^{6}\right ) \]